A note on edge-colourings avoiding rainbow K4 and monochromatic Km
Abstract
We study the mixed Ramsey number maxR(n,Km,Kr), defined as the maximum number of colours in an edge-colouring of the complete graph Kn, such that Kn has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,Km,K4) <= n3/22m for all m >= 3. Further, we discuss a possible way to improve their lower bound on maxR(n,K4,K4) based on incidence graphs of finite projective planes.
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